Nonlinear Resonance Research Articles

Forced resonance of a buckled beam flexibly restrained at the inner point

Restraints along slender structures, such as pipes conveying fluid and beams, play important roles in enhancing their bending stiffness. The current work investigates the influence of the midpoint restraint on the buckled axial force and dynamic characteristics for the first time. A continuous model is introduced by treating the midpoint restraint force as a concentrated force using Dirac function. The influence of the midpoint restraint is discussed based on modal expansion, and a partitioned model is proposed for validation. The conclusion is that the continuous model has good accuracy. Based on this model, it finds out two stable buckled states changing with the midpoint stiffness. The first stable buckled state occurs under weak midpoint restraint and has a cambered shape. The zero- and double frequency components occur in symmetric modal shapes and the response is softened along the whole beam. However, in the second stable buckled state, it turns into a whole sine function shape. The zero- and double frequency components just occur in the second modal shape, and the soften phenomenon doesn’t occur at the midpoint. With increasing excitation amplitude, a bubble appears on the primary resonance peak while another resonance peak emerges nearby it. This work explains the influence of midpoint stiffness on nonlinear resonance and enhances the study of the nonlinear behavior of buckled slender structures.

Controllable nonreciprocal phonon laser in a hybrid photonic molecule based on directional quantum squeezing.

Here, a scheme for a controllable nonreciprocal phonon laser is proposed in a hybrid photonic molecule system consisting of a whispering-gallery mode (WGM) optomechanical resonator and a χ(2)-nonlinear WGM resonator, by directionally quantum squeezing one of two coupled resonator modes. The directional quantum squeezing results in a chiral photon interaction between the resonators and a frequency shift of the squeezed resonator mode with respect to the unsqueezed bare mode. We show that the directional quantum squeezing can modify the effective optomechanical coupling in the optomechanical resonator, and analyze the impacts of driving direction and squeezing extent on the phonon laser action in detail. Our analytical and numerical results indicate that the controllable nonreciprocal phonon laser action can be effectively realized in this system. The proposed scheme uses an all-optical and chip-compatible approach without spinning resonators, which may be more beneficial for integrating and packaging of the system on a chip. Our proposal may provide a new route to realize integratable phonon devices for on-chip nonreciprocal phonon manipulations, which may be used in chiral quantum acoustics, topological phononics, and acoustical information processing.

Nonlinear combined resonance of axially moving conical shells under interaction between transverse and parametric modes

The purpose of present paper is to establish a conical shell model for exploring combined resonance response of axially moving graphene platelets reinforced metal foams (GPLRMF) conical shells subjected to coupled external and parametric excitations in thermal environments. The Reddy's higher order shear deformation theory (HSDT) is applied to derive the stress-strain relationship, and the governing equations of the shell structure are obtained using Hamilton's principle. The displacements and boundary conditions are characterized by a set of displacement shape functions with orthogonal properties. Then, the combined resonant for the conical shell are solved, where the method of varying amplitude (MVA) is adopted for discretizing and formulating the nonlinear governing motion equations. The correctness of the steady-state approximate solutions is verified by conducting comparative analysis, where the existing valuable literature is served as reference. In addition, the effects of initial phase angle, damping interference, material property, axially moving velocity, temperature change, coning angle as well as external excitation amplitude on the vibrational mechanism are analyzed, and the jumping phenomenon, bifurcation behavior, and chaotic characteristics are discussed in detail.

Nonlinear Dynamics and Combination Resonance of a Flexible Turbine Blade with Contact and Friction of Shrouds

Flexible shrouded blades are commonly adopted in the last stages of steam turbines where complicated dynamical behavior can be induced by dry friction force generated on contacting interfaces between adjacent shrouds and the geometric nonlinearity due to the structural flexibility of the blades. In this paper, combination resonance caused by contact and friction forces generated on shroud interfaces is investigated, which is a concurrence of 1:3 internal resonance involving the first and second modes in the flapwise direction and the primary resonance of the first flapwise mode. The stiffness and damping at the contact interface are obtained by linearizing the contact and friction forces between shrouds through the harmonic balance method. The vibrating blade is modeled as a beam with a concentrated mass of which the responses under the combination resonance are solved through the multiple-scale method. Sensitivities of response with respect to the angle of shrouds, contact stiffness and rotation speed are illustrated, and the influences of these parameters on the periodicity and amplitudes of steady responses are demonstrated. The parametric regions where the combination resonance occurs are pointed out. Finally, parametric analyses are presented to show how the amplitude–frequency relation of the multiple-scale solutions under the combination resonance vary with detuning and design parameters. The present research provides a designing basis for improving the dynamic performance of flexible shrouded blades and suppressing vibrations of blades by adjusting structural parameters in practical engineering.

Nonlinear supersonic flutter of a composite panel backed by an acoustic cavity with finite-amplitude sound waves

In this paper, an implicit partitioned fluid-structure-acoustic interaction method is presented for predicting nonlinear aeroelastic response of an elastic composite laminated panel backed by an acoustic cavity and subject to a supersonic flow. The method is formulated in an arbitrary Lagrangian-Eulerian framework based on the unsteady Navier-Stokes equations for the supersonic flow, a geometrical nonlinear dynamics model for the composite laminated panel, and the nonlinear Westervelt equation for the finite-amplitude sound waves in cavity. A theoretical fluid-structure-acoustic model of the supersonic panel is proposed for validating the present numerical method, in which the nonlinear structural model of the panel, the first-order piston aerodynamics model of the supersonic flow and the nonlinear sound wave model of the cavity are coupled in a monolithic manner. It is found that cavity acoustic resonance can appear in the case a vibration mode of the panel coalescing with an acoustic mode of the cavity. Several new physical features of nonlinear acoustic resonance have been discovered, such as the presence of shock wave with high pressure gradients travelling along the resonant cavity in axial direction and the presence of higher-order harmonic wave components, which emphasize a nonlinear acoustic wave model of the cavity must be used to fully characterize these phenomena. In addition, the nonlinearity of the standing wave is quantitatively related to the acoustic modes by means of a proper orthogonal decomposition analysis.

Nonlinear combined harmonic resonances of composite cylindrical shells operating in hygro-thermo-electro-magneto-mechanical fields

By considering the effects of the hygro-thermo-electro-magnetic environments, von Karman nonlinear terms, and multi-harmonic excitations, a coupled nonlinear vibration modeling of composite cylindrical shells comprising a carbon nanotube-reinforced composite (CNTRC) core and two piezoelectric/magnetic composite (PEMC) skins is developed, and the nonlinear dynamic behaviors of such cylindrical shells under primary, super/sub-harmonic, and combined resonance states are investigated. In the theoretical modeling process, the effective mechanical properties of the CNTRC core are first determined using the mixing and Schapery laws, and the hygro-thermo-electro-magneto-mechanical constitutive relations of the PEMC skins are then formulated. Within the framework of Reissner-Mindlin shell theory, the Lagrangian of the system containing Green-Lagrange and von Karman nonlinear terms is derived, and solution techniques based on the multiscale method are provided to obtain nonlinear frequencies, dynamic responses, and phases of CNTRC-PEMC cylindrical shells under multi-physics fields. Consequently, comparison studies are conducted to validate the correctness of the proposed model from different aspects. Based on this, various resonance and chaos behaviors of such structures subjected to multiple load components are revealed and compared, and the influences of environmental factors and structure composition on the amplitude-frequency curves, time-history responses, and phase planes are explored, with several recommendations and findings being drawn.

Application of nonlinear kerr effect 2D PhC-based ring resonators for design all optical 4-bit adder

Application of nonlinear kerr effect 2D PhC-based ring resonators for design all optical 4-bit adder

Nonlinear resonance in oscillatory systems with decaying perturbations

Nonlinear resonance in oscillatory systems with decaying perturbations

Generalized Analysis of Systems Based on Two Nonlinear Resonators

Generalized Analysis of Systems Based on Two Nonlinear Resonators

Self-powered wireless structural sensors for long-term monitoring of bridges

The present work introduces the design of a self-powered wireless static and dynamic structural monitoring system developed by Wisepower Srl and deployed on some bridges along the S.S. 675 "Umbro-Laziale" (former "Civitavecchia - Orte" junction) since 2018. The most significant aspect of the developed monitoring system lies in the fact that it consists of a wireless network of MEMS accelerometers (noise spectral density in all axes: 22.5 μg/√Hz, and sensitivity: 3.9 μg/LSB) powered by energy harvesting systems, which can convert environmental energy (i.e. vibrations and light) into electrical energy collected in batteries, so extending the life and minimizing the maintenance cycles of the devices. Vibration-based energy harvesting is achieved through a non-linear resonator, which utilizes a wider spectrum of frequencies compared to traditional linear vibration energy harvesters owing to its non-linear dynamics. The efficiency of the system is further enhanced by the combination of solar panels, overcoming the limitations associated with traditional wiring or batteries, widening the lifespan of the electronics, and reducing the maintenance costs. The paper reports the analysis method and the data elaboration of some ambient acceleration records, validating the system for natural frequencies identification.

All optical Gray-to-BCD code converter using photonic crystals

Abstract An all optical Gray-to-binary coded decimal (BCD) converter was proposed using nonlinear ring resonators inside square lattice photonic crystals. The proposed structure works based on threshold switching inside nonlinear resonators. The other advantage of the proposed structure is reducing its time response delay. The mean amount of normalized power at the output ports for logic 1 is about 90 %. Also the delay time is about 1 ps.

Effects of multi motion responses and incident-wave height on the gap resonances in a moonpool

The hydrodynamic characteristics of a free-floating moonpool encountering the gap resonances are investigated based on the constrained interpolation profile method in numerical wave tank. This paper mainly concentrates on the influences of the moonpool's motion responses and the incident-wave height on the gap resonances in the free-floating moonpool. Numerical results demonstrated that the heave response significantly changes the frequency and the magnitude of the linear gap resonance, while the roll motion influences more on the vertical wave loads and the wave responses in the fluid field. The heave and roll response of the free-floating moonpool are generally independent. Moreover, the magnitudes of the nonlinear gap resonances have the tendency of catching up and exceeding the linear gap resonance as the incident-wave height increasing for the free-floating moonpool, which is the consequence of the higher-order harmonics driven by the nonlinear processes and the linear secondary resonant region induced by the heave response. Based on the wavelet transform, it could be observed that the amplified harmonic component usually takes more time to be fully developed than other harmonic components during the development of its corresponding nonlinear gap resonance.

On the steady-state interfacial waves with two-dimensional type-A double exact resonance

Steady-state interfacial waves under two-dimensional (2D) type-A exact triad resonance and other related resonances are researched in a two-layer liquid model with a free surface in contact with air. Five groups (groups 1–5) of convergent series solutions are achieved via the homotopy analysis method. It is found that the phenomenon of double exact resonance could exist in periodic interfacial waves if physical parameters correspond to the intersection of two exact resonance curves. The double exact resonance considered here contains a 2D type-A triad resonance and an other resonance. Under the 2D type-A exact triad resonance, the other resonant triad could obviously enlarge or reduce the wave amplitudes and energy proportions of primary and resonant components. Nevertheless, other resonant quartet, quintet, sextet, and septet all produce no influence on interfacial waves when the 2D type-A exact triad resonance occurs. The above-mentioned results indicate that in the neighborhood of the double exact triad resonance, small perturbations of wave vector of a primary component can cause huge changes on wave profiles of free surface and interface, wave amplitude spectrum, and energy distribution of internal waves in real ocean. In addition, the closer the interfacial waves are to the double exact triad resonance, the more possible energy combinations exist in the wave system, and the greater the number of steady-state interfacial wave solutions. All of this should deepen our understanding of nonlinear resonance interactions in short-crested internal waves.

Нелинейные колебания токонесущей струны в магнитном поле

The nonlinear properties of a string must be taken into account when creating string systems that are used in transducers and musical instruments. The forced oscillations of an electrically conductive string in a magnetic field are described by a nonlinear integro-differential equation. The amplitude-frequency characteristics of nonlinear resonances of the first and second orders are obtained. The damping factor of the first-mode oscillations is determined from the optimal coincidence of the experimental and theoretical amplitude-frequency characteristic graphs. The experimental amplitude-frequency characteristics for the first-order resonance is obtained using a laboratory setup with a copper string 0.4 mm in diameter and 0.57 m long, having a certain initial tension. The cumulative damping factor of the first mode is obtained, which takes into account all the dissipative factors of a real string system. Low-frequency thermoparametric oscillations of the second mode are observed in experiments with a string. It follows from the analysis of the Mathieu equation that parametric resonance arises even at very low amplitudes of the thermal tension of a string, and it must be taken into account in the frequency analysis of oscillations of string systems. The authors are grateful to Tomilin V.A. for his help in performing the calculations and graphic work. The authors declare that they have no conflict of interests.

An all-optical 1*2 de-multiplexer based on two-dimensional nonlinear photonic crystal ring resonators

An all-optical 1*2 de-multiplexer based on two-dimensional nonlinear photonic crystal ring resonators

A nonlinear damped metamaterial: Wideband attenuation with nonlinear bandgap and modal dissipation

In this paper, we incorporate the effect of nonlinear damping with the concept of locally resonant metamaterials to enable vibration attenuation beyond the conventional bandgap range. The proposed design combines a linear host cantilever beam and periodically distributed inertia amplifiers as nonlinear local resonators. The geometric nonlinearity induced by the inertia amplifiers causes an amplitude-dependent nonlinear damping effect. Through the implementation of both modal superposition and numerical harmonic methods with Alternating Frequency Time and numerical continuation techniques, the finite nonlinear metamaterial is accurately modeled. The resulting nonlinear frequency response reveals the bandgap is both amplitude-dependent and broadened. Furthermore, the nonlinear interaction between the local resonators and the mode shapes of the host beam is discussed, which leads to efficient modal frequency dissipation ability. The theoretical results are validated experimentally. By embedding the nonlinear damping effect into locally resonant metamaterials, wideband and shock wave attenuation of the proposed metamaterial is achieved, which opens new possibilities for versatile metamaterials beyond the conventional bandgap ranges of their linear counterparts.

All-optical simultaneous OR and NAND gates using photonic crystal ring resonator and Kerr effect

All-optical simultaneous OR and NAND gates design is proposed using a simple nonlinear photonic crystal ring resonator based on the chalcogenide glass (Ag20As32Se48) material which is one of the promising materials for all-optical devices. The structure consists of an octagonal ring resonator between two waveguides, which uses the switching threshold mechanism based on the Kerr effect to perform two simultaneous logic gates functions. The plane wave expansion (PWE) method is used to obtain the band diagram in the proposed structure, and the two-dimensional finite difference time domain (2D-FDTD) method is used in the simulation to evaluate the performance of the proposed design. The resonance wavelength is 1551.3 nm, with a high transmission and coupling efficiency of about 100%. The proposed optical OR and NAND logic gates have high contrast ratios of 21.15 dB and 28.19 dB, respectively, with a quality factor of 596.65. The operating power intensity of the proposed structure is 1 kW μm−2, and the threshold power intensity is obtained at 2.8 kW μm−2. The proposed gates provide transmitted power of not less than 0.5. The size of the structure is 20.5 μm × 18.17 μm. The proposed structure is compact, works on low operational power intensity, and has the ability for dense integration. The simplicity and small size of the structure make it easy to fabricate for future integration in all-optical circuits.

Energy and synchronization between two neurons with nonlinear coupling.

Consensus and synchronous firing in neural activities are relative to the physical properties of synaptic connections. For coupled neural circuits, the physical properties of coupling channels control the synchronization stability, and transient period for keeping energy diversity. Linear variable coupling results from voltage coupling via linear resistor by consuming certain Joule heat, and electric synapse coupling between neurons derives from gap junction connection under special electrophysiological condition. In this work, a voltage-controlled electric component with quadratic relation in the i-v (current-voltage) is used to connect two neural circuits composed of two variables. The energy function obtained by using Helmholtz theorem is consistent with the Hamilton energy function converted from the field energy in the neural circuit. Chaotic signals are encoded to approach a mixed signal within certain frequency band, and then its amplitude is adjusted to excite the neuron for detecting possible occurrence of nonlinear resonance. External stimuli are changed to trigger different firing modes, and nonlinear coupling activates changeable coupling intensity. It is confirmed that nonlinear coupling behaves functional regulation as hybrid synapse, and the synchronization transition between neurons can be controlled for reaching possible energy balance. The nonlinear coupling is helpful to keep energy diversity and prevent synchronous bursting because positive and negative feedback is switched with time. As a result, complete synchronization is suppressed and phase lock is controlled between neurons with energy diversity.

Nonlinear forced vibrations of functionally graded three-phase composite cylindrical shell subjected to aerodynamic forces, external excitations and hygrothermal environment

In this paper, the new functionally graded three-phase composite cylindrical shell is assumed as a common structure in the carrier rocket in the future, and we creatively study the nonlinear forced vibration of this cylindrical shell considering the interaction of different factors in the complex operating environment, including the aerodynamic forces, external excitations, and hygrothermal environment. Based on the first-order shear deformation theory, Von-Karman geometric nonlinear theory, and Hamilton's principle, we derive the nonlinear partial differential equations of motion of the functionally graded three-phase composite cylindrical shell. Considering the axisymmetry of the perfect circular shell, there is a 1:1 internal resonance between the conjugate modes of this cylindrical shell. On this basis, the nonlinear forced vibration of the cylindrical shell is investigated by a combination of Galerkin's method and the pseudo-arc length continuation method. Matcont toolbox can directly solve the ordinary differential equations to obtain the nonlinear frequency response curves. The method can effectively obtain both stable and unstable solutions, avoiding the mathematical difficulties encountered in the formulation process, and facilitating the study of the effects of parametric variables on the resonance response in complex environments. The results show that the variation of material parameters and the complex environment have important effects on the nonlinear resonance response of functionally graded three-phase composite cylindrical shell.

Quantum squeezing-induced quantum entanglement and EPR steering in a coupled optomechanical system.

We propose a theoretical project in which quantum squeezing induces quantum entanglement and Einstein-Podolsky-Rosen steering in a coupled whispering-gallery-mode optomechanical system. Through pumping the χ(2)-nonlinear resonator with the phase matching condition, the generated squeezed resonator mode and the mechanical mode of the optomechanical resonator can generate strong quantum entanglement and EPR steering, where the squeezing of the nonlinear resonator plays the vital role. The transitions from zero entanglement to strong entanglement and one-way steering to two-way steering can be realized by adjusting the system parameters appropriately. The photon-photon entanglement and steering between the two resonators can also be obtained by deducing the amplitude of the driving laser. Our project does not need an extraordinarily squeezed field, and it is convenient to manipulate and provides a novel and flexible avenue for diverse applications in quantum technology dependent on both optomechanical and photon-photon entanglement and steering.